The geodesic flow for discrete groups of infinite volume

Abstract:

Let $\Gamma$ be a discrete group acting in the unit ball $B$ of euclidean $n$-space and $T(B)$ the unit tangent space of $B$. We define the geodesic flow ${g_t}$ on the quotient space $\Omega = T(B)/\Gamma$ and show that for discrete groups of infinite volume the flow is of zero type—namely, for measurable subsets $A,B$ of $\Omega$ which are of finite measure, ${\lim _{t \to \infty }}{g_t}(A) \cap B = 0$. Using this result, we give a new and elementary proof of the fact that for a discrete group of infinite volume, $N(r) = o(V\{ x:\left | x \right | < r\} )$ as $r \to 1$, where $N(r)$ is the orbital counting function and $V$ denotes hyperbolic volume.